Abstract
We revisit the predictions for the pseudoscalar-photon transition form factors in bottom-up and top-down holographic QCD models which only use the pion decay constant and the $\rho$ meson mass as input. We find remarkable agreement with the available experimental data for the single-virtual $\pi^0$ form factor that have recently been extended to lower momenta by BESIII, down to 0.3 GeV$^2$. The bottom-up models moreover turn out to be roughly consistent with recent experimental results obtained by BaBar for the double-virtual $\eta'$ form factor at large momenta as well as with a recent lattice extrapolation for the double-virtual $\pi^0$ form factor. Calculating the pion pole contribution to the hadronic light-by-light scattering in the anomalous magnetic moment of the muon, we find that the bottom-up models in question span the range $a_\mu^{\pi^0}=6.1(4)\cdot 10^{-10}, which is somewhat lower than estimated previously by approximating these holographic predictions through simple interpolators, and in remarkably good agreement with recent results based on a dispersive approach or lattice simulations.
Highlights
The current world average for the experimental value of the anomalous magnetic moment of the muon [1,2], aμ 1⁄4 ðgμ − 2Þ=2, which is dominated by the final result of the E821 Collaboration at Brookhaven National Laboratory obtained 15 years ago, reads [3]aeμxp 1⁄4 ð11659209.1 Æ 6.3Þ × 10−10; ð1Þ with new experiments at FERMILAB and J-PARC aiming at an even more precise determination
Given are simple extrapolations to aημ and aημ0 using the same function KðQ21; Q22Þ obtained in all the holographic models in the chiral limit, but rescaling Fð0; 0Þ according to the central experimental values quoted in [16] [Fηð0; 0Þ=Fπ0ð0; 0Þ 1⁄4 0.2736=0.2725 and Fη0 ð0; 0Þ= Fπ0ð0; 0Þ 1⁄4 0.3412=0.2725] and using the physical masses of η and η0 in the 2-loop integral for aημ;η0.5 (See, e.g., Refs. [47,48] for a discussion of the effects of finite quark masses on pseudoscalar transition form factor (TFF).) Judging from the fitting parameter Λ obtained in Ref. [16], the bottom-up holographic results for KðQ21; Q22Þ match quite well the single-virtual data for η, but for η0 a 10% higher Λ was found
At the smallest values of Q2 this seems perfectly consistent with the preliminary data from BESIII, but for all Q2 > 0.5 GeV2 the experimental data are underestimated, so that the result for aπμ0;SS of 4.8 × 10−10 could perhaps be taken as a lower limit to which positive contributions necessary to make contact with the correct short-distance behavior need to be added
Summary
The current world average for the experimental value of the anomalous magnetic moment of the muon [1,2], aμ 1⁄4 ðgμ − 2Þ=2, which is dominated by the final result of the E821 Collaboration at Brookhaven National Laboratory obtained 15 years ago, reads [3]. [21] to include one of the holographic predictions for the curvature parameters in the TFF does not permit one to fit the full UV behavior of the holographic models, missing in particular the Brodsky-Lepage constraint [27], which is respected qualitatively in all bottomup models This turns out to lead to an overestimation of the single-virtual TFF and, as a consequence, gives a result for aμ that is about 10% higher. The bottom-up-holographic results span the range aπμ0 1⁄4 5.9ð2Þ × 10−10, which turns out to be well in line with the recent result obtained with the DRV interpolator fitted to the world data for π0 [16] and with the lattice results of Ref. The bottom-up-holographic results span the range aπμ0 1⁄4 5.9ð2Þ × 10−10, which turns out to be well in line with the recent result obtained with the DRV interpolator fitted to the world data for π0 [16] and with the lattice results of Ref. [25]
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